Spatio-temporal data are common in practice. Such data often have
complex structures that are difficult to describe and estimate. To
overcome this difficulty, the `R`

package
**SpTe2M** has been developed to estimate the underlying
spatio-temporal mean and covariance structures. This package also
includes tools of online process monitoring to detect change-points in a
spatio-temporal process over time. More specifically, it implements the
nonparametric spatio-temporal data modeling methods described in Yang
and Qiu (2018, 2019, and 2022), as well as the online spatio-temporal
process monitoring methods discussed in Qiu and Yang (2021 and 2023) and
Yang and Qiu (2020). In this vignette, we will demonstrate the main
functions in **SpTe2M** by using the Florida influenza-like
illness (ILI) data, which is a built-in dataset of the package.

The Florida ILI dataset, named as `ili_dat`

in
**SpTe2M**, contains daily ILI incidence rates at 67
Florida counties during years 2012-2014. The observed IIL incidence
rates was collected by the Electronic Surveillance System for the Early
Notification of Community-based Epidemics of the Florida Department of
Health. In addition to observations of the variable `Rate`

(i.e., ILI incidence rate), the dataset also includes observations of
the following 7 variables: `County`

, `Date`

,
`Lat`

(i.e., latitude of the geometric center of a county),
`Long`

(i.e., longitude of the geometric center of a county),
`Time`

, `Temp`

(i.e., air temperature), and
`RH`

(i.e., relative humidity). First of all, let us install
and load the package **SpTe2M** and take a quick look at
the ILI data.

First, we run the following `R`

code to install the
package from CRAN and load the package in `R`

:

Then, we read the built-in dataset `ili_dat`

and use the
`R`

function `head()`

to display its first 6
rows.

```
data(ili_dat)
head(ili_dat)
#> County Date Lat Long Time Rate Temp RH
#> 1 alachua 1/1/2012 29.65320 -82.32440 0.00273224 8.020372e-06 60.5 78.6
#> 2 baker 1/1/2012 30.27236 -82.21351 0.00273224 0.000000e+00 62.0 78.6
#> 3 bay 1/1/2012 30.16190 -85.65297 0.00273224 3.532404e-05 61.5 78.6
#> 4 bradford 1/1/2012 29.96893 -82.12255 0.00273224 0.000000e+00 61.0 78.6
#> 5 brevard 1/1/2012 28.70765 -80.89049 0.00273224 0.000000e+00 68.0 78.6
#> 6 broward 1/1/2012 26.05192 -80.14526 0.00273224 2.527846e-05 71.0 78.6
```

Next, we make some geographic maps to investigate the spatio-temporal
patterns of the ILI incidence. To do this, we first combine
`ili_dat`

with the map shape files included in the packages
**maps** and **mapproj** and then create maps
by using the function `ggplot()`

in the package
**ggplot2**.

```
library(maps)
library(ggplot2)
library(mapproj)
# turn shape files in the maps packages into a data frame
FL <- map_data('county','florida')
names(FL)[6] <- 'County'
# Only plot maps on Jun 15 and Dec 15
subdat <- subset(ili_dat,Date%in%c('6/15/2012','6/15/2013','6/15/2014',
'12/15/2012','12/15/2013','12/15/2014'))
subdat$Date <- factor(subdat$Date,levels=c('6/15/2012','6/15/2013','6/15/2014',
'12/15/2012','12/15/2013','12/15/2014'))
# merge ILI data with map data
mydat <- merge(FL,subdat)
# make maps using ggplot
maps <- ggplot(data=mydat,aes(long,lat,group=group,fill=Rate))+geom_polygon()+
facet_wrap(~Date,ncol=3)+geom_path(colour='grey10',lwd=0.5)+
scale_fill_gradient2('',low='cyan',mid='white',high='navy',
guide='colorbar',limits=c(0,0.0003),na.value='orange',
breaks=c(0,0.0001,0.0002,0.0003),
labels=c('0','1e-4','2e-4','3e-4'))+
guides(fill=guide_colorbar(barwidth=25,barheight=1,direction='horizontal'),
cex=1)+theme_bw(base_size=15)+xlab('longitude')+ylab('latitude')+
theme(legend.position='bottom',
axis.ticks=element_blank(),
line=element_blank(),
axis.text=element_blank(),
panel.border=element_rect(color='black',linewidth=1.2),
axis.line=element_line(colour='black'),
legend.margin=margin(-10,0,0,0),
legend.box.margin=margin(0,0,0,0))
# display the maps
maps
```

From the maps, it can be seen that ILI incidence rates have obvious
seasonal patterns with more ILI cases in the winters and fewer ILI cases
in the summers. Moreover, there seems to be an unusual pattern of ILI
incidence rates in the winter of 2014 because the incidence rates on
12/15/2014 are much higher than those on 12/15/2012 and 12/15/2013.
Next, we appply the package **SpTe2M** to this dataset to
estimate the spatio-temporal mean and covariance structures and monitor
the ILI incidence sequentially over time.

In this part, we estimate the spatio-temporal mean and covariance
structures in year 2013 by using the functions
`spte_meanest()`

and `spte_covest()`

in
**SpTe2M**. To this end, we first use the following code to
extract the observed ILI data in year 2013.

```
n <- 365; m <- 67; N1 <- (n+1)*m; N2 <- n*m
# extract the observed ILI data in year 2013
ili13 <- ili_dat[(N1+1):(N1+N2),]
y13 <- ili13$Rate; st13 <- ili13[,c('Lat','Long','Time')]
```

After obtaining the ILI data in year 2013, the spatio-temporal mean
structure can be estimated by applying the function
`spte_meanest()`

to the extracted data.

Note that we donâ€™t specify the arguments `ht`

and
`hs`

when using the function `spte_meanest()`

. In
such a case, the two bandwidths `ht`

and `hs`

would be selected automatically by the modified cross-validation
procedure (cf., Yang and Qiu 2018).

To visually check whether the estimated means describe the observed ILI data well, we use the code below to plot the estimated means for the 4 Florida counties Broward, Lake, Pinellas, and Seminole, along with the observed ILI incidence rates at the 4 counties.

```
mu <- mu.est$muhat; mu <- t(matrix(mu,m,n))
obs <- t(matrix(y13,m,n))
ids <- c(6,34,52,57) # IDs for Broward, Lake, Pinellas, Seminole
par(mfrow=c(2,2),mgp=c(2.4,1,0))
par(mar=c(3.5,3.5,1.5,0.5))
plot(1:365,mu[,ids[1]],type='l',lty=1,lwd=1.5,xaxt='n',
ylim=c(0,8e-5),main='Broward',xlab='Time',ylab='Incidence rate',
cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
points(1:365,obs[,ids[1]],cex=1)
axis(1,cex.axis=1.2,at=c(1+c(1,62,123,184,245,306, 367)),
label=c('Jan','Mar','May','July','Sep','Nov','Jan'))
par(mar=c(3.5,3,1.5,1))
plot(1:365,mu[,ids[2]],type='l',lty=1,lwd=1.5,xaxt='n',
ylim=c(0,8e-5),main='Lake',xlab='Time',ylab='',
cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
points(1:365,obs[,ids[2]],cex=1)
axis(1,cex.axis=1.2,at=c(1+c(1,62,123,184,245,306, 367)),
label=c('Jan','Mar','May','July','Sep','Nov','Jan'))
par(mar=c(3.5,3.5,1.5,0.5))
plot(1:365,mu[,ids[3]],type='l',lty=1,lwd=1.5,xaxt='n',
ylim=c(0,8e-5),main='Pinellas',xlab='Time',ylab='Incidence rate',
cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
points(1:365,obs[,ids[3]],cex=1)
axis(1,cex.axis=1.2,at=c(1+c(1,62,123,184,245,306, 367)),
label=c('Jan','Mar','May','July','Sep','Nov','Jan'))
par(mar=c(3.5,3,1.5,1))
plot(1:365,mu[,ids[4]],type='l',lty=1,lwd=1.5,xaxt='n',
ylim=c(0,8e-5),main='Seminole',xlab='Time',ylab=' ',
cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
points(1:365,obs[,ids[4]],cex=1)
axis(1.2,at=c(1+c(1,62,123,184,245,306, 367)),
label=c('Jan','Mar','May','July','Sep','Nov','Jan'))
```

It can be seen from the figure that the estimated spatio-temporal
mean structure can well describe the longitudinal pattern of the
observed ILI incidence rates. Thus, the function
`spte_meanest()`

provides a reliable tool for estimating the
spatio-temporal mean function of the ILI data.

To estimate the spatio-temporal covariance, we can use the following code:

In the above command, we donâ€™t specify the arguments `gt`

and `gs`

. Then, the two bandwidths `gt`

and
`gs`

would be determined automatically by minimizing the mean
squared prediction error, as discussed in Yang and Qiu (2019).

In this part, we discuss how to monitor the observed ILI data by
using the functions `sptemnt_cusum()`

,
`sptemnt_ewmac()`

and `sptemnt_ewsl()`

, which
correspond to the CUSUM chart (Yang and Qiu 2020), the EWMAC chart (Qiu
and Yang 2021) and the EWSL chart (Qiu and Yang 2023), respectively.

Notice that the construction of the CUSUM chart (cf., Yang and Qiu 2020) consists of 2 main steps. The first step is to model an in-control (IC) spatio-temporal dataset and estimate the regular spatio-temporal pattern of the IC process. Then, in the second step, the control limit of the chart is first determined by the block bootstrap procedure (cf., Yang and Qiu 2020) and the chart can then be used for online process monitoring. In this example, we use the data in years 2012 and 2013 as the IC data for setting up the chart. These IC observations are divided into two parts: the IC data in year 2013 are used for estimating the mean and covariance functions of the IC spatio-temporal data, and the data in year 2012 are used for determining the control limit of the chart. After that, the chart starts to monitor the ILI incidence rates from the beginning of year 2014.

```
y <- ili_dat$Rate; st <- ili_dat[,c('Lat','Long','Time')]
# specify the argument type
# data with type=IC1 are used to determine the control limit
# data with type=IC2 are used to estimate the regular pattern
# data with type =Mnt are used for sequential process monitoring
type <- rep(c('IC1','IC2','Mnt'),c(N1,N2,N2))
```

By using the code below, we can monitor the ILI incidence rates in year 2014 by the CUSUM chart:

In this example, the bandwidths `ht`

and `hs`

for estimating the mean function are chosen based on the modified
cross-validation procedure (cf., Yang and Qiu 2018), which can be
implemented by using the function `mod_cv()`

in
**SpTe2M**, and the selected bandwidths for `ht`

and `hs`

are 0.05 and 6.50, respectively. Regarding the
bandwidths `gt`

and `gs`

for estimating the
covariance function, they are determined by the mean square prediction
error criterion (cf., Yang and Qiu 2019). After running the function
`cv_mspe()`

, they are chosen to be 0.25 and 1.50,
respectively, in this example.

After obtaining the charting statistics from the above code, we can plot the charting statistic values versus the observation times by using the following code:

```
cstat <- ili.cusum$cstat; cl <- ili.cusum$cl
par(mfrow=c(1,2),mgp=c(2.1,1,0))
# plot the CUSUM chart
par(mar=c(3.5,3.5,1.5,0.5))
plot(1:n,cstat,type="l",xlab="Time",xaxt="n",ylab="Charting statistic",
main='CUSUM',cex=0.6,lwd=1.5,cex.lab=0.7,
cex.axis=0.7,cex.main=0.8)
abline(h=cl,lty=2,lwd=1.5)
axis(1,cex.axis=0.7,at=c(1,59,121,182,243,304,365),
labe= c('Jan','Mar','May','July','Sep','Nov','Jan'))
# plot the zoom-in part
par(mar=c(3.5,3,1.5,1))
plot(259:305,cstat[259:305],type="b",xlab="Time",xaxt="n",ylab=" ",
main='First signal time: 10/16/2014',cex=0.6,lwd=1.5,cex.lab=0.7,
cex.axis=0.7,cex.main=0.8)
abline(h=cl,lty=2,lwd=1.5)
axis(1,cex.axis=0.7,at=c(259,274,289,304),
label=c('Sep 15','Sep 30','Oct 15','Oct 31'))
```

The resulting plot is shown in the left panel of the above figure. To better perceive charting statistic values around its first signal time 10/16/2014, the charting statistic values during the time period from 09/15/2014 to 10/31/2014 are presented in the right panel of the figure.

In the ILI dataset, there are observations of the two covariates
`Temp`

and `RH`

available. It is our belief that
the performance of a control chart can be improved by using the
information in the covariates. To take advantage of the covariate
information, we can use the EWMAC chart (Qiu and Yang 2021) for process
monitoring, which can be implemented by using the function
`sptemnt_ewmac()`

. The following code is an example about how
to use the function `sptemnt_ewmac()`

to monitor the ILI
incidence rates in 2014:

```
x <- as.matrix(ili_dat[,c('Temp','RH')]) # the covariates
ili.ewmac <- sptemnt_ewmac(y,x,st,type,ht=0.05,hs=6.5,gt=0.25,gs=1.5)
```

After running the above code, we use the code below to plot the EWMAC chart, as well as its zoom-in part during the time period from 09/15/2014 to 10/31/2014.

```
cstat <- ili.ewmac$cstat; cl <- ili.ewmac$cl
par(mfrow=c(1,2),mgp=c(2.1,1,0))
# plot the EWMAC chart
par(mar=c(3.5,3.5,1.5,0.5))
plot(1:n,cstat,type="l",xlab="Time",xaxt="n",ylab="Charting statistic",
main='EWMAC',cex=0.6,lwd=1.5,cex.lab=0.7,
cex.axis=0.7,cex.main=0.8)
abline(h=cl,lty=2,lwd=1.5)
axis(1,cex.axis=0.7,at=c(1,59,121,182,243,304,365),
labe= c('Jan','Mar','May','July','Sep','Nov','Jan'))
# plot its zoom-in part
par(mar=c(3.5,3,1.5,1))
plot(259:305,cstat[259:305],type="b",xlab="Time",xaxt="n",ylab=" ",
main='First signal time: 9/23/2014',cex=0.6,lwd=1.5,cex.lab=0.7,
cex.axis=0.7,cex.main=0.8)
abline(h=cl,lty=2,lwd=1.5)
axis(1,cex.axis=0.7,at=c(259,274,289,304),
label=c('Sep 15','Sep 30','Oct 15','Oct 31'))
```

From the figure, it is clear that the EWMAC chart gives its first signal on 09/23/2014, which is 23 days earlier than the first signal of the CUSUM chart. So, it is beneficial to use the covariate information when monitoring the ILI incidence rates.

In many applications, anomalies in spatio-temporal processes start in some small regions that are spatially clustered. To take into account this spatial feature, the EWSL chart was developed for spatio-temporal process monitoring, which is effective in cases when anomalies occur in spatially clustered regions.

Next, we demonstrate the application of the EWSL chart (cf., Qiu and Yang 2023) by using the function sptemnt_ewsl(). To this end, let us use the code as follows:

The plot of the EWSL chart and its zoom-in part during the period from 09/15/2014 to 10/31/2014 can be generated by the code given below.

```
cstat <- ili.ewsl$cstat; cl <- ili.ewsl$cl
par(mfrow=c(1,2),mgp=c(2.1,1,0))
# plot the EWSL chart
par(mar=c(3.5,3.5,1.5,0.5))
plot(1:n,cstat,type="l",xlab="Time",xaxt="n",ylab="Charting statistic",
main='EWSL',cex=0.6,lwd=1.5,cex.lab=0.7,
cex.axis=0.7,cex.main=0.8)
abline(h=cl,lty=2,lwd=1.5)
axis(1,cex.axis=0.8,at=c(1,59,121,182,243,304,365),
labe= c('Jan','Mar','May','July','Sep','Nov','Jan'))
# plot its zoom-in part
par(mar=c(3.5,3,1.5,1))
plot(259:305,cstat[259:305],type="b",xlab="Time",xaxt="n",ylab=" ",
main='First signal time: 10/6/2014',cex=0.6,lwd=1.5,cex.lab=0.7,
cex.axis=0.7,cex.main=0.8)
abline(h=cl,lty=2,lwd=1.5)
axis(1,cex.axis=0.7,at=c(259,274,289,304),
label=c('Sep 15','Sep 30','Oct 15','Oct 31'))
```

It can be seen from the figure that the EWSL chart gives its first signal on 10/6/2014, which is 10 days earlier than that of the CUSUM chart.

Qiu, P. and Yang, K. (2021). Effective Disease Surveillance by Using Covariate Information.

*Statistics in Medicine*,**40**, 5725-5745.Qiu, P. and Yang, K. (2023). Spatio-Temporal Process Monitoring Using Exponentially Weighted Spatial LASSO.

*Journal of Quality Technology*,**55**, 163-180.Yang, K. and Qiu, P. (2018). Spatio-Temporal Incidence Rate Data Analysis by Nonparametric Regression.

*Statistics in Medicine*,**37**, 2094-2107.Yang, K. and Qiu, P. (2019). Nonparametric Estimation of the Spatio-Temporal Covariance Structure.

*Statistics in Medicine*,**38**, 4555-4565.Yang, K. and Qiu, P. (2020). Online Sequential Monitoring of Spatio-Temporal Disease Incidence Rates.

*IISE Transactions*,**52**, 1218-1233.Yang, K. and Qiu, P. (2022). A Three-Step Local Smoothing Approach for Estimating the Mean and Covariance Functions of Spatio-Temporal Data.

*Annals of the Institute of Statistical Mathematics*,**74**, 49-68.